Graphs with the second and third maximum Wiener index over the 2-vertex connected graphs
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Wiener index, defined as the sum of distances between all unordered pairs of vertices, is one of the most popular molecular descriptors. It is well known that among 2-vertex connected graphs on n> 3 vertices, the cycle C_n attains the maximum value of Wiener index. We show that the second maximum graph is obtained from C_n by introducing a new edge that connects two vertices at distance two on the cycle if n 6. If n> 11, the third maximum graph is obtained from a 4-cycle by connecting opposite vertices by a path of length n-3. We completely describe also the situation for n< 10.
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